Voronoi diagrams and applications Prof. Ramin Zabih

2 Administrivia  Last quiz: Thursday 11/15  Prelim 3: Thursday 11/29 (last lecture)  A6 is due Friday 11/30 (LDOC)  Final projects due Friday 12/7  Course evals! –This will count towards your grade! We get a list of students who didn’t fill it out, and will contact you

3 Final projects  All the proposals were fine  Due Friday, Dec 7, 4-5PM –Pizza will be provided! –A few “I survived CS100R” T-shirts  Other CS faculty will likely come by –There may also be photographers, etc.

4 Image differencing Figure from Tim Morris

5 Applying image differencing  By thresholding the difference image, we can figure out which pixels belong to the new object –I.e., the Coke or Pepsi can  These pixels will have some colors –They are points in color space  More generally, Coke pixels will tend to form a group (“cluster”) that is largely red –Pepsi will be largely blue  We can then compute a few model pixels

6 Voronoi diagrams  Divide our space into regions, where each region reg(P) consists of the points closest to a labeled point P –This is a Voronoi diagram –Long history (Descartes, 1644)

7 Impossible algorithms, redux  There are no O(n) sorting algorithms –More precisely, none based on comparisons  You can use convex hull to sort –By placing the points on a parabola –So, is there an O(n) convex hull algorithm?  You can use Voronoi diagrams to compute a convex hull –So, is there an O(n) Voronoi diagram algorithm? You can also use 3D convex hull to compute a 2D Voronoi diagram

8 Using Voronoi diagrams  Two obvious questions: –How can we efficiently create it? –How can we use it, once we’ve got it?  A Voronoi diagram divides the space into Voronoi cells, reg(P) for some P  If reg(P) is a strange shape, hard to figure out if the query is inside reg(P) –Fortunately, as the picture suggests, Voronoi cells have simple shapes

9 Computing reg(P)  Consider some other labeled point Q  Points might be in reg(P) if they are closer to P than to Q –I.e., they are in a polygon (half-plane)  reg(P) is the intersection of N-1 polygons –There are faster ways to compute it

10 Voronoi cell properties  The polygons whose intersection is reg(P) have another important property –They are convex!  The intersection of two convex shapes is also a convex shape

11 Voronoi query lookup  Given a Voronoi diagram and a query point, how do we tell which cell a query falls into? (I.e., solve the 1-NN problem)  We can project down to the x -axis every point in the Voronoi diagram –This gives us a bunch of “slabs” –We can find which slab our query is in by using binary search –Within a slab, we can find the Voronoi cell using binary search –Unfortunately, this is pretty messy

12 Example of slabs Figure from H. Alt “The Nearest Neighbor”

13 A Voronoi application  Image compression: generate a terse representation of the image by allowing small errors to be introduced  Simple method: vector quantization (VQ) –Video applications: Cinepak, Indeo, etc. Also used for audio (Ogg Vorbis) –In a color image, each pixel has a 24-bit number associated with it (the colors) –We can generate a “code book” with, say, 2^8 entries, and use this instead of the colors

14 Example output (Matlab)

15 Triangulations  Suppose that we want to analyze a curved surface, such as a wing or a vase –We can approximate it by a lot of small low- order polygons, especially triangle –This is tremendously important for, e.g., building planes, bridges, or computer graphics Finite Element Method!  Want to build a triangulation out of fat triangles, not skinny ones –Using the Voronoi diagram, we can generate a high-quality triangulation!

16 Delaunay triangulation  Connect two input points if they share an edge in the Voronoi diagram aunay.html  This maximizes the smallest angle  The circumcircle of a triangle passes through all 3 of its vertices –Not the same as the bounding circle –In the Delaunay triangulation, the circumcircles contain no other points

17 Applications of triangulation